Congrats! As long as you are aware of your surroundings and use common sense (i.e. don't walk through dark side streets with your head phones blasting in your ears), you'll be fine.
That being said, there's a good number of housing options around the immediate Columbia area for sublets/lease. I know the 100-110 streets between Broadway and Amsterdam has options and also 122-125 streets between Bway and Amsterdam. You can snag some of the roomshares at pretty decent prices. When I was searching I was looking for below $900 and eventually found an apartment for $850. I chose that over getting a less expensive apartment ($600-750) in Washington Heights or Queens because I dislike commuting and like the option of going home to retrieve things, eat lunch, etc. I probably could've saved $100-200 on rent, but if you figure in expenses that comes with commuting i.e. unlimited metro card, time spent traveling, I thought it was worth it for me in the end.